In a dress rehearsal for the opening weekend of the Aviva Premiership Welsh made the faster start as Tellwright found space out wide before racing around to dot down under the posts.
And the winger was on the hand again with his second of the evening, collecting a wide pass from Aaron Myers to go over on the left.
Leicester could have folded but instead they fought back with Tom Bellak twisting and turning his way to the line for his second try of the evening as the Exiles led 12-5.
But Welsh had the final say in the first half - the impressive Tellwright completing his hat-trick as he and Myers again combined to send the winger under the posts for a 19-5 half-time lead.
Tellwright was not done there - picking up where he had left off in the first half as he added try number four early in the second half.
The prolific winger took a well-earned rest but that didn't slow down the Exiles as Gerhard Wessells and Ben Seru went over in quick succession to stretch the lead out to 40-5.
Leicester clearly had nothing left to give and another break and quick hands saw Seru stopped just short but Rhodri McAtee went off the back of a ruck to add a seventh.
And there was just enough time for Seru to add his second of the match, a simple step doing for the flagging Leicester defence as he went over for a second of the game.